l-independence for Compatible Systems of (mod l) Representations

Let K be a number field. For any system of semisimple mod l Galois representations {\phi_l:Gal_K->GL_N(F_l)} arising from \'etale cohomology, there exists a finite normal extension L of K such that if we denote \phi_l(Gal_K) and \phi_l(Gal_L) by respectively \Gamma_l and \gamma_l for all l, and let S_l be the F_l-semisimple subgroup of GL_N associated to \gamma_l (or \Gamma_l) by Nori [No87] for all sufficiently large l, then the following statements hold for all sufficiently large l: A(i) The formal character of S_l->GL_N is independent of l and is equal to the formal character of the tautological representation of the derived group of the identity component of the monodromy group of the corresponding semi-simplified l-adic Galois representation. A(ii) The non-cyclic composition factors of \gamma_l and S_l(F_l) are identical. Therefore, the composition factors of \gamma_l are finite simple groups of Lie type of characteristic l and cyclic groups. B(i) The total l-rank rk_l\Gamma_l of \Gamma_l is equal to the rank of S_l and is therefore independent of l. B(ii) The A_n-type l-rank rk_l^{A_n}\Gamma_l of \Gamma_l for n belonging to N\{1,2,3,4,5,7,8} and the parity of (rk_l^{A_4}\Gamma_l)/4 are independent of l.